Since momentum modes are conjugate or `dual' to spacetime coordinates \(x^{i}\) , the consideration of momentum and winding
modes at the same level suggests the introduction of a new set of coordinates \(\tilde{x}_{i}\)  `dual' to winding.
This doubling of coordinates allows to formulate a field theory that incorporates the degrees of freedom associated to
winding without necessarily representing strings, and such that T-duality is a manifest symmetry of the theory.
This scheme is called Double Field Theory (DFT) since the pioneering work [1]}
(see also e.g. [2]} for a review), and represents a T-duality covariant formulation of supergravity that incorporates
stringy aspects not present in the usual treatment of supergravity.
Although the doubling of space coordinates should only affect the compact dimensions, it is useful in practice to duplicate all
space coordinates; in this way one ends up with a doubled space, which in the simplest case is a product \(T\times \tilde{T}\) 
and has global coordinates \((x^{i},\tilde{x}_{i})\) ,
where \(T\)  and \(\tilde{T}\)  are \(d\) -dimensional flat tori that are said to be dual from each other.
This `extended spacetime' is intended to be the target space of the fundamental string,
and the physical fieldsIn the simplest formulation of DFT,
the physical fields correspond to the massless sector of closed strings, namely a metric tensor,
a 2-form field (the Kalb-Ramond field) and a scalar field (the dilaton).
are organized in representations of the T-duality group \({\rm O}(d,d)\) .
Furthermore, from the field content of the theory
one deduces a notion of generalized diffeomorphisms, that are infinitesimally generated
by a generalized Lie derivative or `D-bracket'.
Consistency conditions (i.e. closure of the algebra of generalized diffeomorphisms) impose a restriction on the fields.
A particular solution to this restriction is the so-called `section condition' or `strong constraint',
where the fields are forced to depend on only half of the doubled coordinates.
This half is determined as a subspace that is maximally isotropic (or totally null) with respect to the
\({\rm O}(d,d)\)  metric \(\eta ={\rm d}x^{i}\otimes {\rm d}\tilde{x}_{i}+{\rm d}\tilde{x}_{i}\otimes {\rm d}x^{i}\) 
[3]}, i.e. a \(d\) -dimensional subspace \(S\) 
such thatThe metric \(\eta \)  involved in (REF ) does not
represent the gravitational field; it is part of the `kinematics' of the theory.
The gravitational field is unified with the 2-form field into a `generalized metric', which codifies the dynamics;
this will not play a role in the present work.
\(\eta |_{S}=0.\) 
